What is the binomial options pricing model?
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It is a model that employs numbers to approximate the value of options by modeling potential price movements in a straightforward time period.
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What is the Binomial Option Pricing Model?
By Bajaj Broking Team
February 20, 2025
8 mins read
Derivatives
Synopsis:
In this blog, we will discuss the binomial option pricing model and how it works. We will cover the key assumptions that go with this model. And we will talk about its advantages and disadvantages as well.
Mathematics often plays a major role in investment. It allows experts to come up with theories and formulate numerical models for calculations and analysis. These calculations give us certain metrics that help us get an idea of complex trends that may not make sense without numerical data. One such model is the Binomial Option Pricing model. This one focuses on trying to represent price trends in numerical form.
What is the Binomial Option Pricing Model?
The Binomial Option Pricing Model was developed in the late 1970s to give analysts a flexible and intuitive way to carry out option valuation. This model values options by creating simulations. The potential paths that the price of an asset could take, during the life period of the option are simulated for analysis. It breaks down the life period (which ends at the expiry date of the option), into discrete intervals. And at the end of each interval the price is expected to move up or down based on certain factors. This creates a binomial tree of possible price pathways. The star trait of this model is that this can even handle special options with features like early exercise. Most other models struggle to work with such complications.
Understanding the Binomial Option Pricing Model
The key components of the Binomial Option Pricing Model are as follows.
1. Binomial Tree Construction
The Binomial Option Pricing model breaks the time to expiration into many discrete intervals of time.
At every interval, the price of the asset has the potential to move either up or down.
This is the binomial tree, a treelike structure of all possible price paths with time.
The model accommodates the simulation of potential price movements at different times until expiration.
2. Risk-Neutral Valuation
The Binomial Option Pricing model applies the risk-neutral valuation concept, and investors are risk-neutral.
This implies that the expected rate of return on the asset is the risk-free rate.
Risk-neutral probabilities are used in the up move and the down move to estimate an expected payoff.
Option future values are discounted at the risk-free rate to obtain the current value.
3. Valuation of Options
The valuation starts at the terminal nodes of the binomial tree where we approximate the intrinsic value of the option.
The Binomial Option Pricing model works backwards through the tree, calculating the present value of the anticipated payoffs at each node.
The present value is a weighted average of the option values immediately available in the future discounted by the risk-free rate.
This is done until it is at the origin, and it provides the present value of the option.
4. Early Exercise and Decision Nodes
The Binomial Option Pricing model takes into account American options, which allow the option to be exercised earlier before it expires.
In each point where a choice must be made, the model takes into account whether it is optimal to exercise now versus exercising later.
If exercising now is optimal, then early exercising is optimal, and the model delivers that value at the time.
This renders the binomial model better than the Black-Scholes model for American option pricing.
5. Convergence to the Black-Scholes Model
The larger the number of time steps in the binomial tree, the more the results of the model approximate the results of the Black-Scholes model.
This convergence indicates that the binomial approach remains accurate and valid as time intervals become smaller.
Discreteness of the binomial model offers a framework for analyzing continuous-time models like Black-Scholes.
Flexibility of the binomial model allows it to be applied to pricing a broad array of options, including those with early exercise features.
Key Assumptions of the Binomial Option Pricing Model
The Binomial Pricing Model is founded on some significant assumptions that simplify the valuation while ensuring that the option prices are properly estimated. The assumptions constitute the mathematical model that allows analysts and traders to estimate the fair value of an option.
1. Discrete Time
The Binomial Option Pricing model considers the fact that time moves in discrete steps.
At each step, the price of the underlying asset either increases or decreases by a certain percentage.
The model simplifies the computation of option prices at different points in time.
2. No Arbitrage
The Binomial Option Pricing model follows the no-arbitrage principle that guarantees no risk-free profit can be made.
This implies the price of the option should be in sync with the underlying asset price so that there is no price inefficiency taken advantage of by traders.
3. Two Possible Outcomes
Each step of the binomial tree can have only two price movements: up or down.
This is easier to manage and consistent with the manner in which prices move in real life.
4. Constant Volatility
The Binomial Option Pricing model considers the volatility of the asset to be constant throughout the life of the option.
In real life, volatility can change due to market conditions, but the assumption helps in the creation of a structured pricing model.
5. No Dividends
The simple binomial model assumes that the underlying asset does not pay dividends during the life of the option.
When dividends are paid, adjustments are made for their impact on the asset price.
Understanding the Binomial Options Calculations
To arrive at the approximate fair value of an option, the Binomial Option Pricing Model uses a systematic step-by-step calculation process to precisely estimate option prices.
Calculation of Up and Down Factors
Firstly, the upward movement factor (u) and the downward movement factor (d) are the two possible price movements of an asset in each time segment. These are calculated from the asset's volatility and the length of each time segment. With these factors figured out, the model simulates real price movements in the market so that traders can forecast possible price movements.
Risk-Neutral Probability
Next, comes the risk-neutral probability (p) which is an essential ingredient in the model to calculate the expected payoffs of the options. It is obtained with the assumption that investors are risk-neutral; i.e., the return on the asset is the same as the risk-free rate. Multiplying the probability with the possible upward and downward directions, the model generates an unbiased estimate of the option. With this we finally move on to actual option valuation.
Option Valuation at Final Nodes
On the lowest node of the binomial tree, which is the expiry date, the model calculates if the option is in the money or out of the money. If the option is in the money, then its payoff is calculated by looking at the discrepancy between the asset and the strike price. It is an extremely important step in calculating the option value at expiry. The process doesn’t end here though.
Discounting Future Values
Having calculated the option values at the terminal nodes, these are then discounted back along the binomial tree. With the risk-free rate as a basis, the model determines the present value of the option by taking into account the expected payoffs from the following nodes. This is repeated until the starting node is reached, finally determining the current fair value of the option.
Example: Binomial Option Pricing in Practice
To illustrate how the binomial model works, let’s go through a simple example.
Constructing the Binomial Price Tree
Assume a stock is currently trading at ₹100.
The stock can either increase by 10% (u = 1.10) or decrease by 10% (d = 0.90) in each time step.
Over two time steps, we construct the following binomial tree:
Calculating Option Prices at Final Nodes
Suppose we are pricing a European call option with a strike price of ₹100.
At expiration, the option’s payoff is:
Calculating Today’s Option Price
Using the risk-neutral probabilities and discounting process, we work backward to determine the present value of the option.
After applying these calculations, we get an estimated option price at time t = 0.
This step-by-step approach demonstrates how the binomial model helps in pricing options dynamically.
Advantages and Disadvantages of the Binomial Option Pricing Model
Advantages
Flexibility: The model allows for American options, which can be exercised before maturity.
Step-by-step method: Price movements can be examined at various points in time by dealers.
Adjustability: Dividend payments and other real-world considerations can be added into the model.
Accuracy of valuation: The model converges towards the Black-Scholes model as time steps increase.
Disadvantages
Computationally intense: Additional steps in the binomial tree involve intricate calculations.
Assumptions may not be valid: Real-world volatility and dividend payments may not be as the model assumes.
Not for all options: Certain exotic options may need other valuation techniques.
Conclusion
The Binomial Option Pricing Model is a robust and versatile model for option pricing, particularly for early-exercisable options. Its sequential nature provides a clear-cut partition of possible price movements, and it is an indispensable model in financial markets.
However, even though the model is widely used, it is important to understand its assumptions and limitations. The size of the binomial tree steps has a significant impact on accuracy, and traders must ensure they apply realistic market conditions. Despite its complexity, the binomial model is a fundamental pricing technique in the world of options trading.
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Disclaimer: Investments in the securities market are subject to market risk, read all related documents carefully before investing.
This content is for educational purposes only. Securities quoted are exemplary and not recommendatory.
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Frequently Asked Questions
What is the binomial options pricing tool used for?
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It assists traders and analysts in determining the fair value of call and put options based on potential price movements of the underlying asset.
What is the two-state binomial options pricing model?
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This implies that at every time step, the asset price can move in two directions: up or down.
What are the three steps involved in the binomial tree options pricing method?
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Constructing the binomial price tree.
Computing option values at the terminal points.
Working backward to determine the current option price.
What do "u" and "d" represent in the binomial options pricing model?
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"u" is the factor for the upward movement, and "d" is the factor for the downward movement employed to compute asset price movements.
What is the binomial options pricing model with two periods?
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This is a model where price movements occur over two time steps, resulting in a three-level binomial tree for option valuation.
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